← #10573988
Subgroups — #10573988
Order 512.
4584 conjugacy classes of subgroups.
The subgroup lattice has 261 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #53389 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #53386 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #53385 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #53261 | C2 | 1 | 21 | yes | maximal, complemented |
| 5 | #53240 | C2 | 1 | 7 | yes | maximal, complemented |
| 6 | #53189 | C2 | 1 | 21 | yes | maximal, complemented |
| 7 | #53158 | C2 | 1 | 7 | yes | maximal, complemented |
| 8 | #2867 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | #2866 | C22 | 1 | 3 | yes | complemented |
| 10 | #2865 | C22 | 1 | 3 | yes | complemented |
| 11 | #2854 | C22 | 1 | 21 | yes | complemented |
| 12 | #2852 | C22 | 1 | 7 | yes | complemented |
| 13 | #2850 | C22 | 1 | 21 | yes | complemented |
| 14 | #2846 | C22 | 1 | 7 | yes | — |
| 15 | #2803 | C22 | 1 | 21 | yes | complemented |
| 16 | #2801 | C22 | 1 | 7 | yes | complemented |
| 17 | #2785 | C22 | 1 | 7 | yes | complemented |
| 18 | #2783 | C22 | 1 | 21 | yes | complemented |
| 19 | #2778 | C22 | 1 | 21 | yes | complemented |
| 20 | #2773 | C22 | 1 | 21 | yes | complemented |
| 21 | #2772 | C22 | 1 | 21 | yes | complemented |
| 22 | #2759 | C22 | 1 | 21 | yes | complemented |
| 23 | #2756 | C22 | 1 | 21 | yes | complemented |
| 24 | #2753 | C22 | 1 | 21 | yes | complemented |
| 25 | #2736 | C22 | 1 | 7 | yes | — |
| 26 | #2735 | C22 | 1 | 21 | yes | — |
| 27 | #2730 | C22 | 1 | 21 | yes | complemented |
| 28 | #2710 | C22 | 1 | 21 | yes | complemented |
| 29 | #2709 | C22 | 1 | 21 | yes | complemented |
| 30 | #2706 | C22 | 1 | 21 | yes | complemented |
| 31 | #2702 | C22 | 1 | 21 | yes | complemented |
| 32 | #2697 | C22 | 1 | 21 | yes | — |
| 33 | #2696 | C22 | 1 | 21 | yes | complemented |
| 34 | #2685 | C22 | 1 | 21 | yes | — |
| 35 | #2684 | C22 | 1 | 21 | yes | — |
| 36 | #2681 | C22 | 1 | 21 | yes | complemented |
| 37 | #2678 | C22 | 1 | 21 | yes | — |
| 38 | #2678 | C22 | 1 | 21 | yes | — |
| 39 | #2673 | C22 | 1 | 21 | yes | complemented |
| 40 | #2673 | C22 | 1 | 21 | yes | complemented |
| 41 | #2672 | C22 | 1 | 21 | yes | complemented |
| 42 | #2670 | C22 | 1 | 21 | yes | — |
| 43 | #2669 | C22 | 1 | 21 | yes | — |
| 44 | #2668 | C22 | 1 | 21 | yes | — |
| 45 | C26 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×#127 | C23 | 1 | 7 | yes | — |
| 47 | C2×#127 | C23 | 1 | 7 | yes | complemented |
| 48 | C2×#126 | C23 | 1 | 7 | yes | — |
| 49 | C2×#124 | C23 | 1 | 21 | yes | — |
| 50 | C2×#123 | C23 | 1 | 21 | yes | — |
| 51 | C2×#123 | C23 | 1 | 21 | yes | complemented |
| 52 | C2×C22⋊Q8 | C23 | 1 | 21 | yes | complemented |
| 53 | C2×#121 | C23 | 1 | 21 | yes | — |
| 54 | C2×#121 | C23 | 1 | 21 | yes | — |
| 55 | C2×#120 | C23 | 1 | 21 | yes | — |
| 56 | C2×#120 | C23 | 1 | 21 | yes | — |
| 57 | C2×#120 | C23 | 1 | 7 | yes | — |
| 58 | C22×C42 | C23 | 1 | 7 | yes | complemented |
| 59 | #401 | C23 | 1 | 7 | yes | complemented |
| 60 | #400 | C23 | 1 | 21 | yes | complemented |
| 61 | #399 | C23 | 1 | 21 | yes | — |
| 62 | #399 | C23 | 1 | 7 | yes | complemented |
| 63 | #398 | C23 | 1 | 21 | yes | complemented |
| 64 | #397 | C23 | 1 | 21 | yes | — |
| 65 | #397 | C23 | 1 | 21 | yes | — |
| 66 | #397 | C23 | 1 | 21 | yes | — |
| 67 | #397 | C23 | 1 | 21 | yes | — |
| 68 | #396 | C23 | 1 | 21 | yes | — |
| 69 | #396 | C23 | 1 | 21 | yes | — |
| 70 | #396 | C23 | 1 | 7 | yes | — |
| 71 | #395 | C23 | 1 | 7 | yes | complemented |
| 72 | #394 | C23 | 1 | 21 | yes | — |
| 73 | #394 | C23 | 1 | 21 | yes | — |
| 74 | #394 | C23 | 1 | 21 | yes | — |
| 75 | #394 | C23 | 1 | 21 | yes | — |
| 76 | #394 | C23 | 1 | 21 | yes | — |
| 77 | #394 | C23 | 1 | 21 | yes | — |
| 78 | #394 | C23 | 1 | 21 | yes | complemented |
| 79 | #394 | C23 | 1 | 21 | yes | complemented |
| 80 | #393 | C23 | 1 | 21 | yes | — |
| 81 | #393 | C23 | 1 | 7 | yes | — |
| 82 | #393 | C23 | 1 | 7 | yes | complemented |
| 83 | #392 | C23 | 1 | 21 | yes | — |
| 84 | #392 | C23 | 1 | 21 | yes | — |
| 85 | #392 | C23 | 1 | 21 | yes | — |
| 86 | #392 | C23 | 1 | 21 | yes | — |
| 87 | #391 | C23 | 1 | 21 | yes | complemented |
| 88 | #390 | C23 | 1 | 21 | yes | — |
| 89 | #390 | C23 | 1 | 21 | yes | — |
| 90 | #390 | C23 | 1 | 21 | yes | — |
| 91 | #389 | C23 | 1 | 21 | yes | complemented |
| 92 | #389 | C23 | 1 | 21 | yes | complemented |
| 93 | #388 | C23 | 1 | 21 | yes | — |
| 94 | #388 | C23 | 1 | 21 | yes | — |
| 95 | #388 | C23 | 1 | 21 | yes | — |
| 96 | #388 | C23 | 1 | 21 | yes | — |
| 97 | #388 | C23 | 1 | 21 | yes | — |
| 98 | #388 | C23 | 1 | 21 | yes | — |
| 99 | #388 | C23 | 1 | 21 | yes | — |
| 100 | #388 | C23 | 1 | 21 | yes | complemented |
| 101 | #388 | C23 | 1 | 21 | yes | complemented |
| 102 | #388 | C23 | 1 | 21 | yes | complemented |
| 103 | #388 | C23 | 1 | 21 | yes | complemented |
| 104 | #387 | C23 | 1 | 21 | yes | complemented |
| 105 | #387 | C23 | 1 | 21 | yes | complemented |
| 106 | #387 | C23 | 1 | 21 | yes | complemented |
| 107 | #386 | C23 | 1 | 21 | yes | — |
| 108 | #386 | C23 | 1 | 21 | yes | — |
| 109 | #386 | C23 | 1 | 21 | yes | — |
| 110 | #386 | C23 | 1 | 21 | yes | — |
| 111 | #386 | C23 | 1 | 21 | yes | complemented |
| 112 | #386 | C23 | 1 | 21 | yes | complemented |
| 113 | #386 | C23 | 1 | 21 | yes | complemented |
| 114 | #385 | C23 | 1 | 21 | yes | complemented |
| 115 | #383 | C23 | 1 | 7 | yes | complemented |
| 116 | #382 | C23 | 1 | 21 | yes | complemented |
| 117 | #380 | C23 | 1 | 21 | yes | complemented |
| 118 | #379 | C23 | 1 | 21 | yes | — |
| 119 | #379 | C23 | 1 | 7 | yes | — |
| 120 | #379 | C23 | 1 | 21 | yes | — |
| 121 | C43 | C23 | 1 | 1 | yes | characteristic, complemented |
| 122 | C25 | — | 8 | 7 | no | — |
| 123 | C25 | C24 | 1 | 3 | yes | — |
| 124 | C25 | C24 | 1 | 3 | yes | — |
| 125 | C25 | C24 | 1 | 1 | yes | characteristic |
| 126 | C22×Q8 | C24 | 1 | 7 | yes | — |
| 127 | C22×D4 | C24 | 1 | 21 | yes | — |
| 128 | C22×D4 | C24 | 1 | 21 | yes | — |
| 129 | C22×D4 | C24 | 1 | 7 | yes | — |
| 130 | C22×D4 | C24 | 1 | 21 | yes | — |
| 131 | C23×C4 | C24 | 1 | 21 | yes | — |
| 132 | #127 | — | 8 | 7 | no | — |
| 133 | #127 | — | 8 | 7 | no | — |
| 134 | #126 | — | 8 | 7 | no | — |
| 135 | #124 | — | 8 | 21 | no | — |
| 136 | #123 | — | 8 | 21 | no | — |
| 137 | #123 | — | 8 | 21 | no | — |
| 138 | C22⋊Q8 | — | 8 | 21 | no | — |
| 139 | #121 | — | 8 | 21 | no | — |
| 140 | #121 | — | 8 | 21 | no | — |
| 141 | #120 | — | 8 | 21 | no | — |
| 142 | #120 | — | 8 | 7 | no | — |
| 143 | #120 | — | 8 | 21 | no | — |
| 144 | C2×C4⋊C4 | C24 | 1 | 21 | yes | — |
| 145 | C2×C4⋊C4 | C24 | 1 | 21 | yes | — |
| 146 | C2×C4⋊C4 | C24 | 1 | 21 | yes | — |
| 147 | C2×C4⋊C4 | C24 | 1 | 21 | yes | — |
| 148 | C2×C4⋊C4 | C24 | 1 | 21 | yes | — |
| 149 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 150 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 151 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 152 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 153 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 154 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 155 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 156 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 157 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 158 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 159 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 160 | C2×C22⋊C4 | C24 | 1 | 21 | yes | — |
| 161 | C2×C42 | — | 8 | 7 | no | — |
| 162 | C2×C42 | C24 | 1 | 21 | yes | — |
| 163 | C2×C42 | C24 | 1 | 7 | yes | — |
| 164 | C2×C42 | C24 | 1 | 7 | yes | — |
| 165 | #95 | C24 | 1 | 21 | yes | — |
| 166 | #95 | C24 | 1 | 21 | yes | — |
| 167 | #95 | C24 | 1 | 21 | yes | — |
| 168 | #95 | C24 | 1 | 21 | yes | — |
| 169 | #95 | C24 | 1 | 21 | yes | — |
| 170 | #95 | C24 | 1 | 7 | yes | — |
| 171 | #95 | C24 | 1 | 21 | yes | — |
| 172 | #95 | C24 | 1 | 21 | yes | — |
| 173 | C24 | — | 4 | 7 | no | — |
| 174 | C24 | — | 4 | 21 | no | — |
| 175 | C24 | — | 4 | 21 | no | — |
| 176 | C24 | — | 8 | 56 | no | — |
| 177 | C24 | C25 | 1 | 3 | yes | — |
| 178 | C24 | C25 | 1 | 3 | yes | — |
| 179 | C24 | C25 | 1 | 1 | yes | characteristic |
| 180 | C24 | C25 | 1 | 7 | yes | — |
| 181 | C2×Q8 | — | 4 | 21 | no | — |
| 182 | C2×D4 | — | 4 | 21 | no | — |
| 183 | C2×D4 | — | 4 | 21 | no | — |
| 184 | C2×D4 | — | 4 | 21 | no | — |
| 185 | C2×D4 | — | 4 | 21 | no | — |
| 186 | C2×D4 | — | 4 | 21 | no | — |
| 187 | C2×D4 | — | 4 | 21 | no | — |
| 188 | C2×D4 | — | 4 | 21 | no | — |
| 189 | C2×D4 | — | 4 | 21 | no | — |
| 190 | C2×D4 | — | 4 | 21 | no | — |
| 191 | C2×D4 | — | 4 | 21 | no | — |
| 192 | C22×C4 | — | 4 | 21 | no | — |
| 193 | C22×C4 | — | 4 | 21 | no | — |
| 194 | C22×C4 | — | 4 | 21 | no | — |
| 195 | C22×C4 | C25 | 1 | 7 | yes | — |
| 196 | C22×C4 | C25 | 1 | 21 | yes | — |
| 197 | C22×C4 | C25 | 1 | 21 | yes | — |
| 198 | C4⋊C4 | — | 4 | 21 | no | — |
| 199 | C4⋊C4 | — | 4 | 21 | no | — |
| 200 | C4⋊C4 | — | 4 | 21 | no | — |
| 201 | C4⋊C4 | — | 4 | 21 | no | — |
| 202 | C4⋊C4 | — | 4 | 21 | no | — |
| 203 | C22⋊C4 | — | 4 | 21 | no | — |
| 204 | C22⋊C4 | — | 4 | 21 | no | — |
| 205 | C22⋊C4 | — | 4 | 21 | no | — |
| 206 | C22⋊C4 | — | 4 | 21 | no | — |
| 207 | C22⋊C4 | — | 4 | 21 | no | — |
| 208 | C22⋊C4 | — | 4 | 21 | no | — |
| 209 | C22⋊C4 | — | 4 | 21 | no | — |
| 210 | C22⋊C4 | — | 4 | 21 | no | — |
| 211 | C22⋊C4 | — | 4 | 21 | no | — |
| 212 | C22⋊C4 | — | 4 | 21 | no | — |
| 213 | C22⋊C4 | — | 4 | 21 | no | — |
| 214 | C22⋊C4 | — | 4 | 21 | no | — |
| 215 | C42 | — | 4 | 21 | no | — |
| 216 | C42 | — | 4 | 7 | no | — |
| 217 | C42 | — | 4 | 7 | no | — |
| 218 | C23 | — | 2 | 21 | no | — |
| 219 | C23 | — | 2 | 21 | no | — |
| 220 | C23 | — | 2 | 21 | no | — |
| 221 | C23 | — | 2 | 7 | no | — |
| 222 | C23 | — | 2 | 21 | no | — |
| 223 | C23 | — | 2 | 7 | no | — |
| 224 | C23 | — | 8 | 14 | no | — |
| 225 | C23 | — | 8 | 42 | no | — |
| 226 | C23 | — | 8 | 42 | no | — |
| 227 | C23 | — | 8 | 64 | no | — |
| 228 | C23 | C26 | 1 | 1 | yes | characteristic |
| 229 | Q8 | — | 16 | 7 | no | — |
| 230 | D4 | — | 16 | 21 | no | — |
| 231 | D4 | — | 16 | 21 | no | — |
| 232 | D4 | — | 16 | 21 | no | — |
| 233 | D4 | — | 16 | 7 | no | — |
| 234 | C2×C4 | — | 2 | 21 | no | — |
| 235 | C2×C4 | — | 2 | 21 | no | — |
| 236 | C2×C4 | — | 2 | 21 | no | — |
| 237 | C2×C4 | — | 2 | 21 | no | — |
| 238 | C2×C4 | — | 2 | 21 | no | — |
| 239 | C2×C4 | — | 2 | 21 | no | — |
| 240 | C2×C4 | — | 2 | 21 | no | — |
| 241 | C2×C4 | — | 8 | 42 | no | — |
| 242 | C22 | — | 4 | 21 | no | — |
| 243 | C22 | — | 4 | 21 | no | — |
| 244 | C22 | — | 4 | 7 | no | — |
| 245 | C22 | — | 4 | 21 | no | — |
| 246 | C22 | — | 4 | 21 | no | — |
| 247 | C22 | — | 4 | 7 | no | — |
| 248 | C22 | — | 8 | 8 | no | — |
| 249 | C22 | — | 8 | 24 | no | — |
| 250 | C22 | — | 8 | 24 | no | — |
| 251 | C22 | H(7,𝔽2) | 1 | 7 | yes | — |
| 252 | C4 | — | 4 | 21 | no | — |
| 253 | C4 | — | 4 | 21 | no | — |
| 254 | C4 | — | 4 | 7 | no | — |
| 255 | C2 | — | 8 | 7 | no | — |
| 256 | C2 | — | 8 | 3 | no | — |
| 257 | C2 | — | 8 | 3 | no | — |
| 258 | C2 | — | 8 | 1 | no | — |
| 259 | C2 | #62973 | 1 | 7 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta